Question

Determine whether each of the following sets with the defined operations are groups. Motivate your answer. (a) S=Z with multiplication; (b) S={2ⁿ3ᵐ ∣n,m∈Z} with multiplication.

Answer 1

To determine if S=Z with multiplication is a group, we need to check if it satisfies four group axioms: closure, associativity, identity, and inverse. After checking all four axioms, we find that S=Z with multiplication is a group.

Step-by-step explanation:

(a) To determine if S=Z with multiplication is a group, we need to check if it satisfies four group axioms:

1. **Closure**: For any two elements a and b in S, the product ab must also be in S.
2. **Associativity**: For any three elements a, b, and c in S, the equation (ab)c = a(bc) holds.
3. **Identity**: There exists an element e in S such that for any element a in S, ae = ea = a.
4. **Inverse**: For every element a in S, there exists an element a^-1 in S such that aa^-1 = a^-1a = e.

Let's check each axiom for S=Z with multiplication:

1. **Closure**: For any two integers a and b, the product ab is also an integer, so closure is satisfied.
2. Associativity: Multiplication is associative for integers, so associativity is satisfied.
3. **Identity**: The identity element for multiplication in integers is 1, so the identity axiom is satisfied.
4. **Inverse**: For any integer a, the inverse element is 1/a, which is also an integer, so the inverse axiom is satisfied.

Since all four group axioms are satisfied, S=Z with multiplication is a group.